Entangled regression curves growing out of a cracked data tablet, symbolizing flawed data analysis in economic forecasts.

Is Your Data Lying to You? Unmasking Hidden Biases in Time Series Regressions

Theo Raines in Business & Economy December 2025 • 5 min read.

"Discover how weak exogeneity and autocorrelated regressors can lead to flawed conclusions in economic analyses – and how to fix it."

In the dynamic world of economics, researchers frequently rely on time series data to understand and predict market trends. However, the reliability of these analyses hinges on the assumptions made about the data. One of the most common assumptions, known as 'weak exogeneity,' posits that the error term in a regression model is independent of past and present values of the regressors. This allows for errors to correlate with future regressor values, a seemingly benign condition.

However, recent research has uncovered a concerning issue: in time series regressions with many control variables, weak exogeneity can lead to substantial biases. These biases can be so significant that they render the ordinary least squares (OLS) estimator inconsistent, meaning it fails to converge to the true value as the sample size increases. This is especially problematic because OLS is a foundational tool in econometrics.

This article delves into the nature of these biases, explaining why they arise in regressions with numerous regressors and how they can be addressed. We'll explore an innovative bias correction approach that offers improved properties relative to OLS, ensuring more accurate and reliable results. Understanding and mitigating these biases is crucial for anyone working with time series data in economics and related fields.

The Trouble with Weak Exogeneity: How Biases Creep In

Entangled regression curves growing out of a cracked data tablet, symbolizing flawed data analysis in economic forecasts.

The core of the problem lies in the behavior of the OLS design matrix when many regressors are involved. In such cases, the normalized OLS design matrix remains asymptotically random, even with large sample sizes. When only weak exogeneity holds, this randomness becomes correlated with the regression error, creating a bias that undermines the accuracy of the estimates. The magnitude of this bias is directly related to the number of regressors and their average autocorrelation. High autocorrelation means that past values of the regressors are strong predictors of their current values, exacerbating the bias.

Traditional methods assume strict exogeneity which is a condition rarely met due to feedback loops within economic systems, where today's outcomes influence tomorrow's conditions. This violation of strict exogeneity, even in a single period, can set off a chain reaction of biases.

Feedback Loops: The biases stem from feedback effect, where the outcome variable in one period influences the regressors in future periods. This is very common in macroeconomic models.
Number of Regressors: The more regressors included in the analysis, the larger the potential bias.
Autocorrelation: High autocorrelation among the regressors amplifies the effect of weak exogeneity on the bias.

Consider how economic policies are evaluated using time series regressions. If these regressions suffer from biases due to weak exogeneity, policymakers could be making decisions based on flawed information, leading to unintended consequences.

Correcting Course: A New Approach to More Accurate Analysis

To combat these issues, researchers have developed innovative approaches to bias correction. One such method involves creating a new estimator with improved properties relative to OLS. This estimator addresses the bias by mimicking an instrumental variables (IV) estimator, using a 'technical' instrument that is intentionally endogenous. The key is that the future values of the regressors in the instrument induce an endogeneity bias along the same feedback direction as the original OLS bias. By carefully selecting the weights in the linear combination, it's possible to ensure that the bias stemming from the endogenous instrument offsets the bias originating from weak exogeneity. This new estimator is consistent and, after proper normalization, asymptotically Gaussian, providing a reliable method for inference.

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This article is based on research published under:

DOI-LINK: https://doi.org/10.48550/arXiv.2308.08958,

Title: Linear Regression With Weak Exogeneity

Subject: econ.em stat.me

Authors: Anna Mikusheva, Mikkel Sølvsten

Published: 17-08-2023

Everything You Need To Know

What is weak exogeneity, and why is it a problem in time series regressions?

Weak exogeneity is an assumption in time series regressions that states the error term is independent of past and present values of the regressors. The issue arises because weak exogeneity can lead to substantial biases in time series regressions, especially when many control variables are present. These biases make the ordinary least squares (OLS) estimator inconsistent, meaning it doesn't converge to the true value as the sample size grows. The magnitude of the bias is related to the number of regressors and their average autocorrelation.

How does the number of regressors and autocorrelation impact the biases caused by weak exogeneity?

The number of regressors and the degree of autocorrelation play critical roles in exacerbating the biases caused by weak exogeneity. The larger the number of regressors included in the analysis, the larger the potential bias. High autocorrelation means that past values of the regressors strongly predict their current values, which amplifies the bias introduced when weak exogeneity holds. This means that the more complex the model (more regressors) and the more interconnected the data (higher autocorrelation), the more likely and severe the biases become.

Why is strict exogeneity rarely met in economic systems, and what are the implications?

Strict exogeneity, which assumes that the error term is independent of all past, present, and future values of the regressors, is rarely met due to feedback loops inherent in economic systems. These feedback loops mean that today's outcomes can influence tomorrow's conditions. The violation of strict exogeneity, even in a single period, can trigger a chain reaction of biases. For example, a macroeconomic model where the outcome variable influences the regressors in future periods will suffer from these biases.

What are the core issues leading to bias in time series regressions, as explained in the text, and how do they manifest?

The core of the problem lies in the behavior of the OLS design matrix when many regressors are involved and when only weak exogeneity holds. In such instances, the OLS design matrix remains asymptotically random, leading to a correlation between this randomness and the regression error. This correlation creates bias, which undermines the accuracy of the estimates. The key factors amplifying this problem are feedback loops (where outcomes influence regressors in the future), the number of regressors (more regressors increase potential bias), and autocorrelation (high autocorrelation amplifies the bias).

What innovative approach is suggested to correct the biases caused by weak exogeneity, and how does it work?

To combat the biases caused by weak exogeneity, the text mentions a new estimator with improved properties relative to OLS. This estimator addresses the bias by mimicking an instrumental variables (IV) estimator, using a 'technical' instrument that is intentionally endogenous. The instrument leverages future values of the regressors to induce an endogeneity bias that counteracts the original OLS bias. By carefully selecting the weights in the linear combination, the bias from the endogenous instrument offsets the bias from weak exogeneity. The resulting estimator is consistent and asymptotically Gaussian, providing a reliable method for inference.